GENIKEUMENA OLOKLHRWMATA

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Σοφός Αργυριάδης
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1 PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA I GENIKEUMENA OLOKLHRWMATA Anplhrwt c Kjhght c: Dr. Pppˆc G. Alèndroc

2 GENIKEUMENA OLOKLHRWMATA H ènnoi tou orismènou oloklhr mtoc mic sunˆrthshc proüpojètei ìti h sunˆrthsh eðni frgmènh s> èn frgmèno diˆsthm α, β]. Epiplèon sto jemeli dec je rhm tou OloklhrwtikoÔ LogismoÔ upojètoume ìti h sunˆrthsh f eðni suneq c sto α, β]. J melet soume sth sunèqei oloklhr mt mic prgmtik c metblht c t opoð den plhroôn mð ki tic dôo utèc proüpojèseic. Autˆ t oloklhr mt onomˆzonti genikeumèn oloklhr mt (improper integrls). Orismìc: KloÔme genikeumèno olokl rwm to β ìtn isqôei mi pì tic sunj kec: α f() () H oloklhrwtè sunˆrthsh f() èqei èn pepersmèno pl joc shmeðwn suneqeðc sto α, β]. (b) 'En toulˆqiston pì t ìri oloklhr sewc eðni to ˆpeiro. Apì tic sunj kec pou jèsme dikrðnoume tic e c peript seic ditup nontc touc ntðstoiqouc orismoôc. PerÐptwsh h: An h sunˆrthsh f() eðni suneq c α, β), llˆ suneq c sto shmeðo β, tìte orðzoume: β β f() f() () α α () An to ìrio upˆrqei ki eðni prgmtikìc rijmìc, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An to ìrio den upˆrqei, tìte lème ìti to genikeumèno olokl rwm poklðnei den orðzeti. PerÐptwsh h: An h sunˆrthsh f() eðni suneq c (α, β], llˆ suneq c sto shmeðo α, tìte orðzoume: β β f() f() () α α+ () An to ìrio upˆrqei ki eðni prgmtikìc rijmìc, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An to ìrio den upˆrqei, tìte lème ìti to genikeumèno olokl rwm poklðnei den orðzeti.

3 PerÐptwsh 3h: An h f() eðni suneq c α, β] eiroumènou tou shmeðou γ ( pepersmènou pl jouc shmeðwn ξ, ξ,..., ξ ν ) ìpou orðzoume: β α γ (α, β) ξ i (α, β), i,,..., ν] f() γ α β f() + lim f() (3) γ+ (ntist. gi to ξ i ), n t ìri tou deutèrou mèlouc thc (3) upˆrqoun. () An t ìri upˆrqoun, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An èn pì t ìri tou deutèrou mèlouc toulˆqiston den upˆrqei tìte to genikeumèno olokl rwm poklðnei den orðzeti. PerÐptwsh 4h: An h f() eðni suneq c α, u] orðzoume: α f() u lim u + α f() (4) () An to ìrio upˆrqei ki eðni prgmtikìc rijmìc, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An to ìrio den upˆrqei, tìte lème ìti to genikeumèno olokl rwm poklðnei den orðzeti. PerÐptwsh 5h: An h f() eðni suneq c ν, β] orðzoume: β f() β lim ν ν f() (5) () An to ìrio upˆrqei ki eðni prgmtikìc rijmìc, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An to ìrio den upˆrqei, tìte lème ìti to genikeumèno olokl rwm poklðnei den orðzeti. PerÐptwsh 6h: An h f() eðni suneq c ν, u] orðzoume: f() α lim ν ν f() + lim u + u α f() (6) () An t ìri upˆrqoun, tìte lème ìti to genikeumèno olokl rwm sugklðnei ìti èqei ènnoi. (b) An èn pì t ìri tou deutèrou mèlouc toulˆqiston den upˆrqei tìte to genikeumèno olokl rwm poklðnei den orðzeti. 3

4 Krit ri sôgklishc (Ôprhc) genikeumènwn oloklhrwmˆtwn 'Estw h mh rnhtik sunˆrthsh f :, + ) R. Upojètoume ìti h f eðni suneq c sto diˆsthm, + ). To genikeumèno olokl rwm f() sugklðnei ìtn upˆrqei jetikìc rijmìc ϑ (, + ) tètoioc, ste u f() ϑ gi kˆje u, + ). Orismìc: To genikeumèno olokl rwm f() sugklðnei polôtwc, ìtn ki mìno ìtn sugklðnei to olokl rwm f() An to genikeumèno olokl rwm sugklðnei, tìte sugklðnei ki to genikeumèno olokl rwm f() ki mˆlist lhjeôei f() f() f(). 4

5 Krit rio sôgkrishc 'Estw f, g :, + ) R sunrt seic me pedðo orismoô to diˆsthm, + ) ki oloklhr simec se kˆje kleistì upodiˆsthm tou, + ) me R. Upojètoume epðshc ìti Tìte lhjeôoun oi kìloujec idiìthtec: f() g() gi kˆje, + ) i. An sugklðnei to genikeumèno olokl rwm tìte sugklðnei ki to genikeumèno olokl rwm g(), ki j isqôei h nisìtht f() f() g(). ii. An poklðnei to genikeumèno olokl rwm tìte poklðnei ki to genikeumèno olokl rwm f() dhld n tìte j isqôei ki g(), f() + g() +. 5

6 Orikì krit rio sôgkrishc: 'Estw f, g :, + ) R sunrt seic me pedðo orismoô to diˆsthm, + ) ki oloklhr simec se kˆje kleistì upodiˆsthm tou, + ) me R. Upojètoume epðshc ìti f() ki g() > gi kˆje, + ). 'Estw f() lim + g() L R Tìte lhjeôoun oi kìloujec idiìthtec: i. An L ki to genikeumèno olokl rwm g(), sugklðnei, tìte sugklðnei ki to genikeumèno olokl rwm f(). ii. An L (, + ), tìte t genikeumèn oloklhr mt f() ki g() prousiˆzoun thn Ðdi sumperiforˆ wc proc th sôgklish, dhld sugklðnoun ki t dôo oloklhr mt poklðnoun ki t dôo oloklhr mt. iii. An L + ki g() +, tìte ki f() +. 6

7 Prˆdeigm.: 'Estw to olokl rwm: Eetˆsete eˆn sugklðnei. 4. LÔsh H oloklhrwtè sunˆrthsh eðni suneq c gi, ˆr: ki n jèsoume 4 4 t ki dt to ìristo olokl rwm dðnei: 4 ki sunep c: dt 4 t dt t τoξηµt τoξηµ 4 τoξηµ τoξηµ π. 'Ar to olokl rwm sugklðnei ki èqei tim 4 ] τoξηµ π. ] τoξηµ 7

8 Prˆdeigm.: Eetˆsete n to olokl rwm sugklðnei. LÔsh Epeid h eðni suneq c sto èqoume: 'Ar sugklðnei ki èqei tim + ] + ] +.. Prˆdeigm.3: Eetˆsete n sugklðnei to: ln. LÔsh Epeid h oloklhrwtè sunˆrthsh eðni suneq c sto ˆnw ìrio, j èqoume: sunep c sugklðnei. u ln u + ] u u + ln u + ln u + ] ln + ln ln ln d ln 8

9 Prˆdeigm.4: 'Estw to olokl rwm: Eetˆsete eˆn sugklðnei. 3. LÔsh H oloklhrwtè sunˆrthsh: eðni suneq c sto, ìpou (, ), 3 sunep c: lim δ + ] + lim δ + ( ) + ( ) +, dhld to I den èqei ènnoi gitð to ìrio den upˆrqei. +δ ] + lim δ + 3 ] +δ ] + + lim δ + ] ( + δ) ] ( + δ) + ShmeÐwsh: An to I to upologðsoume sn orismèno olokl rwm qwrðc n lˆboume upìyh, ìti gi eðni suneq c j èqoume: 3 ] +. H tim ut tou I eðni lnjsmènh gitð h 3 ki sunep c to I den mhdenðzeti potè sto diˆsthm, ]. 9

10 Prˆdeigm.5: 'Estw to olokl rwm Eetˆsete n sugklðnei ìqi LÔsh Epeid t ìri eðni èqoume: 'Ar sugklðnei. ν + 9 ν lim u + u lim ν τoξϕ3] ν + 3 lim u + τoξϕ3]u 3 lim τoξϕ τoξϕ3ν] + ν 3 lim τoξϕ3u τoξϕ] u + ( π )] + π ] 3 3 π 3.

11 Prˆdeigm.6: N upologðsete to olokl rwm p, ìpou p R. Apˆnthsh. To, p R sugklðnei gi p > ki poklðnei gi p. p Apˆnthsh JewroÔme th sunˆrthsh f :, + ) R pou orðzeti me ton tôpo f() gi kˆje p R. p H sunˆrthsh f eðni suneq c ki epomènwc oloklhr simh sto diˆsthm, t] gi kˆje t >. Gi p èqoume: Epomènwc t p t p ] t p p ( ) p t p t lim t + p p, n p > +, n p <. Gi p, lmbˆnoume t ln ]t ln t. Sunep c t lim t + +. 'Ar to olokl rwm p sugklðnei gi p > ki poklðnei gi p.

12 Prˆdeigm.7: N eetstoôn wc proc th sôgklish t genikeumèn oloklhr mt () sin. ShmeÐwsh. To p, p R sugklðnei gi p > ki poklðnei gi p y y sin y y (b) DikiologeÐste gewmetrikˆ gitð to sin +. Apˆnthsh

13 () sin u sin u + u u + ( cos ) { ] u u ( ) ( cos ) ( cos )} u + { ] u u ( ) u + cos + cos } u u ] cos u + cos cos u + lim u + u cos u + cos lim cos lim u + u cos u u + u cos cos. u cos u u lim u + u + u cos u. cos sugklðnei (Krit rio sôgkrishc) cos sugklðnei cos sugklðnei pìlut 'Ar to sin sugklðnei. cos sugklðnei. 3

14 (b) To sin poklðnei. y y sin y y y y y sin y sin y 4

15 Prˆdeigm.8: N upologðsete ton ìgko tou stereoô ek peristrof c gôrw pì ton -ôn tou qwrðou pou perikleðeti pì thn kmpôlh y ln ki tic eujeðec, y. LUSH H sunˆrthsh f : (, ] R pou orðzeti me f() ln eðni mh rnhtik ki suneq c sto (, ]. 'Omwc h f den eðni frgmènh gi. 'Estw mi sunˆrthsh y f() suneq c sto α, β] ki oi eujeðec α, β, me f(), α, β]. Oi eujeðec, to tìo ÂB thc y f() pou periorðzeti pì tic eujeðec, ki o -ônc orðzoun èn epðpedo qwrðo E. O ìgkoc tou stereoô pou prˆgeti pì thn peristrof tou qwrðou E perð ton -ôn eðni: V π β α f() ]. Sunep c èqoume: V π J eetˆsoume thn Ôprh tou orðou lim (ln ) π lim (ln ). (ln ) ( ln + ln ). 5

16 'Ar V π dhld o zhtoômenoc ìgkoc tou stereoô eðni ShmeÐwsh. (ln ) (ln ) π, V π κ.µ. (ln ) ln + ln. (ln ) () (ln ) (ln ) ] (ln ) ] (ln ) (ln ) ln ln ln ln ln () ln ln { ln ] ln ln ln ln ln ln + ln + ] ln + ln + ln + ln. ln (ln ) (Prgontik Olokl rwsh) (Prgontik Olokl rwsh) } (ln ) ] ] ln lim ln + + (L Hôpitl) (ln ) ( + ) + + ( ). lim ln ln + + ( (L Hôpitl) ln ) ( + ) ln (ln ) ln + + ln lim + lim + ln. ln + 6

17 Prˆdeigm.9: N upologðsete to embdìn tou qwrðou pou perikleðeti pì th grfik prˆstsh thc ki twn sumpt twn thc. y y y y LUSH E 4 4 y 4 lim 4 4 τ.µ. ShmeÐwsh. An to olokl rwm eðni thc morf c: R (, ) b, jètoume: b ηµt, π t π. 7

18 α, β, ηµt, π t π, συνtdt ki ηµ t συνt, opìte: Apì to sq m ηµt συνt dt συνt ηµt dt συνt + C C A - t B ìmwc èqoume: συνt ètsi to olokl rwm I gðneti telikˆ: + C. ] ( ) + lim E 4 lim 4 4 τ.µ. 8

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